Systems and Methods for Characterizing Poroelastic Materials

ABSTRACT

Disclosed herein are systems and methods for characterizing poroelastic materials. Indentation of a poroelastic solid by a spherical-tip tool is analyzed within the framework of Biot&#39;s theory. The present disclosure provides the response of the indentation force as well as the field variables as functions of time when the rigid indenter is loaded instantaneously to a fixed depth. Some embodiments of the present disclosure consider the particular case when the surface of the semi-infinite domain is permeable and under a drained condition. Compressibility of both the fluid and solid phases is taken into account. The solution procedure based on the McNamee-Gibson displacement function method is adopted.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional PatentApplication No. 62/658,840, filed 17 Apr. 2018, the entire contents andsubstance of which is incorporated herein by reference in its entiretyas if fully set forth below.

FIELD OF THE DISCLOSURE

The present disclosure relates generally to characterization of solidmaterials. Particularly, embodiments of the present disclosure relate tosystems and methods for characterizing poroelastic materials.

BACKGROUND

The process of indentation by a rigid tool has been widely studied forits versatility as an experimental technique to probe constitutiveproperties of materials of various kinds across multiple scales.Recently, spherical indentation has been applied to characterizeporoelasticity of fully saturated porous media, such as polymeric gelsand hydrated bones, via either displacement- or force-controlled tests.In a displacement-controlled load relaxation test, the indenter ispressed instantaneously to a fixed depth and held until the indentationforce approaches a horizontal asymptote, whereas in a step force loadingor ramp-hold test, the indentation force is kept constant after reachinga prescribed level. In theory, for a step loading test, if both thesolid and fluid phases can be considered incompressible, elasticconstants can be determined from the early and late time responses,while the hydraulic diffusivity can be obtained form the transientresponse. In the literature, only a few cases of poroelastic contactproblems have been investigated theoretically and have been treated asincompressible. Additionally, the methods provide difficulties insolving the complex equations, such as evaluating integrals with rapidoscillation.

What is needed, therefore, is an improved material characterizationtechnique to take into account the compressibility of both the fluid andsolid phases while overcoming the mathematical difficulties presented bythe prior art. Embodiments of the present disclosure address this needas well as other needs that will become apparent upon reading thedescription below in conjunction with the drawings.

BRIEF SUMMARY OF THE INVENTION

The present invention relates to systems and methods for characterizingporoelastic materials. An exemplary embodiment of the present inventioncan provide a method for characterizing poroelastic materials,comprising: obtaining experimental data for a material, comprising atleast: time data and indentation force data; indicating a firstasymptote of the indentation force data at a corresponding first timefrom the time data; indicating a second asymptote of the indentationforce data at a corresponding second time from the time data; selectinga corresponding master curve from a plurality of theoretical mastercurves based on the ratio of the first and the second asymptotes; andcalculating a value for a property of the material by matching theexperimental data with the corresponding master curve.

In any of the embodiments disclosed herein, the material can be aporoelastic solid.

In any of the embodiments disclosed herein, the obtaining theexperimental data can comprise: indenting, with an indentation tool, theporoelastic solid to a predetermined indentation depth; measuring aforce required to maintain the indentation tool at the indentation depthto obtain the indentation force data; and recording the indentationforce data with respect to the corresponding time data.

In any of the embodiments disclosed herein, the indentation tool cancomprise a rigid smooth sphere.

In any of the embodiments disclosed herein, the rigid sphere can beselected from the group consisting of: permeable indenters andimpermeable indenters.

In any of the embodiments disclosed herein, the drainage condition onthe surface of the poroelastic solid can be selected from the groupconsisting of: a permeable indenter on a fully permeable surface, apermeable/impermeable indenter on a fully impermeable surface, and animpermeable indenter on a fully permeable surface.

In any of the embodiments disclosed herein, the first time from the timedata can be an initial time, wherein the indentation force reaches amaximum.

In any of the embodiments disclosed herein, the second time from thetime data can be an ending time.

In any of the embodiments disclosed herein, the desired materialproperty can be the coefficient of hydraulic diffusion.

In any of the embodiments disclosed herein, the experimental data cancomprise at least a force relaxation behavior.

In any of the embodiments disclosed herein, the determining acorresponding master curve can comprise: calculating a constantcomprising the ratio of the first and the second asymptotes; andselecting a corresponding master curve from a theoretical solutioncorresponding to the value of the constant and the drainage condition ofthe contact surface.

In any of the embodiments disclosed herein, the calculating the value ofthe property of the material can comprise fitting, using a fittingfunction predetermined by the full poroelastic solution, theexperimental data to the corresponding master curve to obtain the valueof the property of the material.

In any of the embodiments disclosed herein, the plurality of mastercurves can be obtained by: establishing one or more governing equations;establishing one or more boundary conditions using Heaviside stepdisplacement loading; transforming the one or more governing equationsusing Hankel transform in the Laplace domain to find the expressions forthe two displacement functions in terms of integrals with threeunknowns; transforming the one or more boundary conditions to theLaplace domain and matching the boundary conditions with the fieldquantities expressed through the displacement functions to obtain threeequations for the three unknowns, which includes one or more Fredholmintegral equations of the second kind; providing alternate integralexpressions for the integral kernels in the Fredholm integral equationsusing one or more modified Struve functions; solving the Fredholmintegral equations using a method of successive substitution;calculating a time domain solution by numerically inverting the solvedLaplace domain equation; and integrating, based on a plurality ofparameters of a specific poroelastic material, the time domain solutionto obtain a plurality of master curves representing force relaxation.

In any of the embodiments disclosed herein, the one or more governingequations can be obtained from one or more McNamee-Gibson displacementfunction methods.

In any of the embodiments disclosed herein, the one or more governingequations can comprise one or more material constants.

In any of the embodiments disclosed herein, the expressions of one ormore field quantities can comprise the displacement function solutionsfrom the one or more governing equations.

Another embodiment of the present disclosure can provide a system forcharacterizing poroelastic materials, comprising: an indentation tool,comprising a rigid smooth sphere; one or more force sensors; one or moreprocessors; and at least one memory storing instructions that whenexecuted by the one or more processors, cause the system to: indent,with the indentation tool, a material to a predetermined indentationdepth; measure, using the one or more force sensors, a force required tomaintain the indentation tool at the indentation depth to obtainindentation force data; record experimental data for the material,comprising at least: indentation force data and time data correspondingto the indentation force data; indicate a first asymptote of theindentation force data at a corresponding first time from the time data;indicate a second asymptote of the indentation force data at acorresponding second time from the time data; select a correspondingmaster curve from a plurality of theoretical master curves based on theratio of the first and the second asymptotes; and calculate a value fora property of the material by matching the experimental data with thecorresponding master curve.

In any of the embodiments disclosed herein, the material can be aporoelastic solid.

In any of the embodiments disclosed herein, the rigid sphere can beselected from the group consisting of: permeable indenters andimpermeable indenters.

In any of the embodiments disclosed herein, the drainage condition onthe surface of the poroelastic solid can be selected from the groupconsisting of: a permeable indenter on a fully permeable surface, apermeable/impermeable indenter on a fully impermeable surface, and animpermeable indenter on a fully permeable surface.

In any of the embodiments disclosed herein, the first time from the timedata can be an initial time, wherein the indentation force reaches amaximum.

In any of the embodiments disclosed herein, the second time from thetime data can be an ending time.

In any of the embodiments disclosed herein, the desired materialproperty can be the coefficient of hydraulic diffusion.

In any of the embodiments disclosed herein, the experimental data cancomprise at least a force relaxation behavior.

In any of the embodiments disclosed herein, the determining acorresponding master curve can comprise: calculating a constantcomprising the ratio of the first and the second asymptotes; andselecting a corresponding master curve from a theoretical solutioncorresponding to the value of the constant and the drainage condition ofthe contact surface.

In any of the embodiments disclosed herein, the calculating the value ofthe property of the material can comprise fitting, using a fittingfunction predetermined by the full poroelastic solution, theexperimental data to the corresponding master curve to obtain the valueof the property of the material.

In any of the embodiments disclosed herein, the plurality of mastercurves can be obtained by storing instructions that when executed by theone or more processors, cause the system to: establish one or moregoverning equations; establish one or more boundary conditions usingHeaviside step displacement loading; transform the one or more governingequations using Hankel transform in the Laplace domain to find theexpressions for the two displacement functions in terms of integralswith three unknowns; transform the one or more boundary conditions tothe Laplace domain and matching the boundary conditions with the fieldquantities expressed through the displacement functions to obtain threeequations for the three unknowns, which includes one or more Fredholmintegral equations of the second kind; provide alternate integralexpressions for the integral kernels in the Fredholm integral equationsusing one or more modified Struve functions; solve the Fredholm integralequations using a method of successive substitution; calculate a timedomain solution by numerically inverting the solved Laplace domainequation; and integrate, based on a plurality of parameters of aspecific poroelastic material, the time domain solution to obtain aplurality of master curves representing force relaxation.

In any of the embodiments disclosed herein, the one or more governingequations can be obtained from one or more McNamee-Gibson displacementfunction methods.

In any of the embodiments disclosed herein, the one or more governingequations can comprise one or more material constants.

In any of the embodiments disclosed herein, the expressions of one ormore field quantities can comprise the displacement function solutionsfrom the one or more governing equations.

These and other aspects of the present invention are described in theDetailed Description of the Invention below and the accompanyingfigures. Other aspects and features of embodiments of the presentinvention will become apparent to those of ordinary skill in the artupon reviewing the following description of specific, exemplaryembodiments of the present invention in concert with the figures. Whilefeatures of the present invention may be discussed relative to certainembodiments and figures, all embodiments of the present invention caninclude one or more of the features discussed herein. Further, while oneor more embodiments may be discussed as having certain advantageousfeatures, one or more of such features may also be used with the variousembodiments of the invention discussed herein. In similar fashion, whileexemplary embodiments may be discussed below as device, system, ormethod embodiments, it is to be understood that such exemplaryembodiments can be implemented in various devices, systems, and methodsof the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of this specification, illustrate multiple embodiments of thepresently disclosed subject matter and serve to explain the principlesof the presently disclosed subject matter. The drawings are not intendedto limit the scope of the presently disclosed subject matter in anymanner.

FIG. 1 is a flowchart of an exemplary embodiment of a method forcharacterizing poroelastic materials;

FIG. 2 is a flowchart of an exemplary embodiment of a method forcharacterizing poroelastic materials;

FIG. 3 is a graph of contact pressure at various times for an exemplaryembodiment of a method for characterizing poroelastic materials;

FIG. 4 is a graph of radial stress at dimensionless times for anexemplary embodiment of a method for characterizing poroelasticmaterials;

FIG. 5 is a graph of distribution of pore pressure at dimensionlesstimes for an exemplary embodiment of a method for characterizingporoelastic materials;

FIG. 6 is a graph of a plurality of master curves for relaxation ofnormalized indentation force obtained from an exemplary embodiment of amethod for characterizing poroelastic materials;

FIG. 7 is a flowchart of an exemplary embodiment of a method forcharacterizing poroelastic materials; and

FIG. 8 is a graph of a comparison between direct numerical integrationof an oscillatory kernel and the alternative expression using anexemplary embodiment of the present disclosure.

DETAILED DESCRIPTION

Although certain embodiments of the disclosure are explained in detail,it is to be understood that other embodiments are contemplated.Accordingly, it is not intended that the disclosure is limited in itsscope to the details of construction and arrangement of components setforth in the following description or illustrated in the drawings. Otherembodiments of the disclosure are capable of being practiced or carriedout in various ways. Also, in describing the embodiments, specificterminology will be resorted to for the sake of clarity. It is intendedthat each term contemplates its broadest meaning as understood by thoseskilled in the art and includes all technical equivalents which operatein a similar manner to accomplish a similar purpose.

Herein, the use of terms such as “having,” “has,” “including,” or“includes” are open-ended and are intended to have the same meaning asterms such as “comprising” or “comprises” and not preclude the presenceof other structure, material, or acts. Similarly, though the use ofterms such as “can” or “may” are intended to be open-ended and toreflect that structure, material, or acts are not necessary, the failureto use such terms is not intended to reflect that structure, material,or acts are essential. To the extent that structure, material, or actsare presently considered to be essential, they are identified as such.

By “comprising” or “containing” or “including” is meant that at leastthe named compound, element, particle, or method step is present in thecomposition or article or method, but does not exclude the presence ofother compounds, materials, particles, method steps, even if the othersuch compounds, material, particles, method steps have the same functionas what is named.

It is also to be understood that the mention of one or more method stepsdoes not preclude the presence of additional method steps or interveningmethod steps between those steps expressly identified.

The components described hereinafter as making up various elements ofthe disclosure are intended to be illustrative and not restrictive. Manysuitable components that would perform the same or similar functions asthe components described herein are intended to be embraced within thescope of the disclosure. Such other components not described herein caninclude, but are not limited to, for example, similar components thatare developed after development of the presently disclosed subjectmatter.

As described above, a problem with current techniques for characterizingporoelastic materials is the mathematical difficulty of solving thecomplex systems of equations for including the compressibility of bothsolids and liquids. Improved solid characterization techniques canvastly improve the design space in industries such as drilling, failureanalysis, geology, soil management, erosion control, and the like.

Indentation of a poroelastic solid by a spherical-tip tool can beanalyzed within the framework of Biot's theory. Embodiments of thepresent disclosure seek the response of the indentation force as well asthe field variables as functions of time when the rigid indenter isloaded instantaneously to a fixed depth. Three particular cases can beconsidered when the drainage condition of the surface of thesemi-infinite domain is one of the following: 1) a permeable indenter ona fully permeable surface (case I or drained case), 2) apermeable/impermeable indenter on a fully impermeable surface (case IIor undrained case), 3) an impermeable indenter on a fully permeablesurface (case III or mixed case). Compressibility of both the fluid andsolid phases can be taken into account.

Theoretical solutions for a poroelastic half space subjected to surfaceloading can be found for a few special cases. The McNamee-Gibsondisplacement function method has generally been employed to solve thisclass of problems. In the earlier literature, the constituents aretreated as incompressible. McNamee and Gibson considered uniformmechanical loading along an infinite strip and over a circular area,respectively. While the McNamee-Gibson displacement function methodprovides a general solution procedure, one of the difficulties insolving this class of problems analytically is in evaluating integralswith oscillatory kernels over an unbounded interval. In the presentdisclosure, it is shown that such issues can be overcome by the use of aseries of special functions.

Problem formulation and the solution procedure are first introduced.Solutions for the contact pressure and relaxation of the indentationforce with time are shown.

In practice, governing equations for an axisymmetric full poroelasticproblem in a half space (z≥0) can be written using the McNamee-Gibsondisplacement functions

and

, namely,

$\begin{matrix}{{\frac{\partial}{\partial t}{\nabla^{2}}} = {c{\nabla^{2}{\nabla^{2}}}}} & (1) \\{{\nabla^{2}\mathcal{F}} = 0} & (2) \\{where} & \; \\{\nabla^{2}{= {{{\frac{\partial^{2}}{\partial r^{2}}{+ \frac{1}{r}}}\frac{\partial}{\partial r}} + \frac{\partial^{2}}{\partial z^{2}}}}} & \; \\{c = \frac{\kappa \; \left( {K + {\frac{4}{3}G}} \right)}{\left\lbrack {\alpha^{2} + {S\left( {K + {\frac{4}{3}G}} \right)}} \right\rbrack \mu}} & \; \\{\alpha = {1 - \frac{K}{K_{s}}}} & \; \\{S = {\frac{n}{K_{f}} + \frac{\alpha - n}{K_{s}}}} & \;\end{matrix}$

The following list of independent material constants can be used todescribe the poroelastic process:

κ—permeability

μ—fluid viscosity

K—skeleton bulk modulus

G—skeleton shear modulus

n—porosity

K_(f)—fluid bulk modulus

K_(s)—solid bulk modulus

Five derived material constants can be used to facilitate thederivation. In addition to the diffusivity coefficient, c, the Biotcoefficient, α, and the storage coefficient, S, two other materialconstants η and ϕ, are defined as follows

$\eta = {\frac{K}{2G} + \frac{2}{3}}$$\varphi = \frac{\alpha^{2} + {S\; \left( {K + {\frac{4}{3}G}} \right)}}{\alpha^{2} + {S\left( {K + {\frac{1}{3}G}} \right)}}$

Field variables such as the stresses, pore pressure and displacementscan be directly expressed using the displacement functions

and

, for example,

$\begin{matrix}{\frac{\sigma_{Z}}{2G} = {{- {\nabla^{2}}} + \frac{\partial^{2}}{\partial z^{2}} - {z\frac{\partial^{2}\mathcal{F}}{\partial z^{2}}} + {\varphi \frac{\partial\mathcal{F}}{\partial z}}}} & (3)\end{matrix}$

where σ_(z) is the vertical stress. Compression positive is adopted forthe sign convention.

The general solutions for Eqs. 1 and 2 can be obtained by a Hankeltransformation in the Laplace domain. After neglecting the termsunbounded at infinite, one can obtain,

$\begin{matrix}{\overset{\_}{} = {\int_{0}^{\infty}{\begin{bmatrix}{{A_{1}{\exp \left( {{- z}\; \xi} \right)}} +} \\{A_{2}{\exp \left( {{- z}\sqrt{\xi^{2} + \lambda}} \right)}}\end{bmatrix}{J_{0}\left( {r\; \xi} \right)}d\; \xi}}} & (4) \\{\overset{\_}{\mathcal{F}} = {\int_{0}^{\infty}{B_{1}{\exp \left( {{- z}\; \xi} \right)}{J_{0}\left( {r\; \xi} \right)}d\; \xi}}} & (5)\end{matrix}$

where Δ=s/c and s is the Laplace transform parameter. A₁, A₂ and B₁ arefunctions of ξ and s to be determined through the boundary conditions.The overbar is used here to denote the functions in the Laplace domain.

Heaviside step displacement loading can be applied to the spherical-tipindenter. Conformity is assumed at the frictionless contact surface. Itcan be shown that if a sphere is pressed to a fixed depth in aporoelastic medium, the contact radius in fact changes with time. It isunclear whether such a problem with a free and moving boundary can besolved analytically. Here, the assumption is made that change in thecontact radius is small and the contact radius a remains fixed and canbe determined from a=√{square root over (Rd)}, where d is the depth ofpenetration and R is the radius of the indenter.

The boundary conditions for the problem can be written in terms ofsurface displacement u_(z), stresses σ_(z), σ_(zr) and pore pressure p:

Case I Case II Case III 0 ≤ r ≤ a⁻$u_{z} = {\left( {d - \frac{r^{2}}{2R}} \right){\mathcal{H}(t)}}$  σ_(zr) = 0, p = 0$u_{z} = {\left( {d - \frac{r^{2}}{2R}} \right){\mathcal{H}(t)}}$  σ_(zr) = 0, q_(z) = 0$u_{z} = {\left( {d - \frac{r^{2}}{2R}} \right){\mathcal{H}(t)}}$  σ_(zr) = 0, q_(z) = 0 r ≥ a₊ σ_(z) = σ_(zr) = 0 p = 0 σ_(z) = σ_(zr) = 0q_(z) = 0 σ_(z) = σ_(zr) = 0, p = 0where

(t) is the Heaviside step function.

Transformation of the non-trivial B.C. in the Laplace domains gives,

0 ≤ r ≤ a⁻${\overset{\_}{u}}_{z} = {\left( {d - \frac{r^{2}}{2R}} \right)s^{- 1}}$

For Case I, matching the poroelastic fields with the boundary conditionsin the Laplace domain can yield the following equations,

A ₁ ξ+A ₂√{square root over (ξ²+λ)}+B ₁(1−ϕ)=0  (6)

A ₂ ηλ+B ₁ξ[ϕ+2η(1−ϕ)]=0  (7)

and a dual integral equation containing only the unknown B₁,

$\begin{matrix}{{{\int_{0}^{\infty}{B_{1}J_{0}\left\{ {r\; \xi} \right)d\; \xi}} = {\left( {s\; \varphi} \right)^{- 1}\left( {\frac{r^{2}}{2R} - d} \right)}}{0 \leq r \leq a_{-}}} & (8) \\{{{\int_{0}^{\infty}{\left\lbrack {1 + {H\left( {s,\xi} \right)}} \right\rbrack \xi \; B_{1}{J_{0}\left( {r\; \xi} \right)}d\; \xi}} = 0}\mspace{14mu} {r \geq a_{+}}{where}{{H\left( {s,\xi} \right)} = {\omega \left( {1 + {2\lambda^{- 1}\xi^{2}} - {2\lambda^{- 1}\xi \sqrt{\xi^{2} + \lambda}}} \right)}}{\omega = \frac{\varphi + {2{\eta \left( {1 - \varphi} \right)}}}{\varphi \left( {{2\eta} - 1} \right)}}} & (9)\end{matrix}$

Constant ω can be expressed explicitly using other material constants

$\begin{matrix}{\omega = \frac{\alpha^{2}\left( {1 - {2v}} \right)}{\alpha^{2} + {2SG\frac{1 - v}{1 - {2v}}}}} & (10)\end{matrix}$

where ν is the drained Poisson's ratio. Since 0≤α≤1, S≥0, G≥0 and0≤ν≤0.5, the theoretical range of ω is [0,1]. If both the fluid andsolid phases are incompressible, ω reduces to a function of ν only,ω=1-2ν.

It has been shown that the solution of a dual integral equation can beexpressed by,

$\begin{matrix}{{B_{1}\xi^{- \frac{1}{2}}} = {\sqrt{\frac{2}{\pi}}{\left( \frac{1}{s\; \varphi \; R} \right)\begin{bmatrix}{{\int_{0}^{a}{{x^{\frac{1}{2}}\left( {x^{2} - {Rd}} \right)}{J_{- \frac{1}{2}}\left( {x\; \xi} \right)}{dx}}} +} \\{\int_{a}^{\infty}{x\; {\overset{\_}{\theta}\left( {s,x} \right)}{J_{- \frac{1}{2}}\left( {x\; \xi} \right)}{dx}}}\end{bmatrix}}}} & (11)\end{matrix}$

where

$J_{- \frac{1}{2}}\left( {x\; \xi} \right)$

is the Bessel function of the first kind of order −½ and θ(s, m)satisfies a Fredholm integral equation of the second kind.

For convenience, the following dimensionless variables are introduced,

x _(*) =x/a r _(*) =r/a z _(*) =z/a

ξ_(*) =ξa s _(*) =λa ² t _(*) =tc/a ²

Denote function θ ₁(s_(*), x_(*)) as the normalized θ(s, x),

${{\overset{¯}{\theta}}_{1}\left( {s_{*},x_{*}} \right)} = {{\overset{¯}{\theta}\left( {s,x} \right)}a^{- \frac{3}{2}}}$

The Fredholm integral equation for θ ₁(s_(*), x_(*)) is,

$\begin{matrix}{{\begin{bmatrix}{{{\overset{¯}{\theta}}_{1}\left( {s_{*},x_{*}} \right)} +} \\{\omega {\int_{1}^{\infty}{{N\left( {s_{*},x_{*},m_{*}} \right)}{{\overset{\_}{\theta}}_{1}\left( {s_{*},m_{*}} \right)}d\; m_{*}}}}\end{bmatrix} = {\omega \; {M\left( {s_{*},x_{*}} \right)}}}{{where},{{N\left( {s_{*},x_{*},m_{*}} \right)} = {m_{*}{\int_{0}^{\infty}{\xi_{*}{H_{1}\left( {s_{*},\xi_{*}} \right)}{J_{- \frac{1}{2}}\left( {x_{*}\xi_{*}} \right)}{J_{- \frac{1}{2}}\left( {m_{*}\xi_{*}} \right)}d\; \xi_{*}}}}}}{{M\left( {s_{*},x_{*}} \right)} = {\int_{0}^{1}{{m_{*}^{- \frac{1}{2}}\left( {1 - m_{*}^{2}} \right)}{N\left( {s_{*},x_{*},m_{*}} \right)}d\; m_{*}}}}{{H_{1}\left( {s_{*},\xi_{*}} \right)} = \left( {1 + {2s_{*}^{- 1}\xi_{*}^{2}} - {2s_{*}^{- 1}\xi_{*}\sqrt{\xi_{*}^{2} + s_{*}}}} \right)}} & (12)\end{matrix}$

Eq. 12 can then be evaluated numerically, where θ ₁(s_(*), x_(*)) is theunknown to be determined. As would be appreciated by one of ordinaryskill in the art, so long as the dimensionless spatial and temporalcoordinates x_(*) and t_(*) (or s_(*) in the Laplace domain) are fixed,θ ₁(s_(*), x_(*)) is influenced only through the derived materialconstant ω.

Function N(s_(*), x_(*), m_(*)) can be evaluated prior to finding thesolution to Eq. 12. Though uniformly convergent, N(s_(*), x_(*), m_(*))has an oscillatory integral kernel over an unbounded interval. As x_(*)and m_(*) become large, the rapidly oscillating integrand could resultin unstable numerical integration. Using current methods known in theart, the integrand can be separated into two parts: one dealing with itsasymptote a s_(*) ⁻¹ξ_(*) ²→∞, the integral of which can be expressed inclosed-form; the other being the difference between the integrand andthe asymptote, which reduces to zero faster and thus can be numericallytreated more effectively. Alternate methods use a function to fitH₁(s_(*), f_(*)). The fitting function is chosen in such a way that itcan be integrated analytically.

In some embodiments of the present disclosure, a different approach overknown methods can be adopted by providing an alternative integralexpression for N(s_(*), x_(*), m_(*)). The expression for N(s_(*),x_(*), m_(*)) can be rewritten using one of the integral representationsof the modified Struve functions, in which the oscillatory nature can beremoved. A comparison between direct numerical integration of theoscillatory kernel and the alternative expression using the modifiedStruve functions for N(s_(*), x_(*), m_(*)) at x_(*)=1, s_(*)=5,000 and500,000 are shown in FIG. 8, wherein m_(*) is a scaled radius on thesurface. The results in both cases are obtained by directly using the“integral” command in MATLAB®, or any other such software able toperform such operations. Direct integration yields highly oscillatoryresults at large s_(*) (small t_(*)) and fails to capture the Diracfunction behavior at m_(*)=x_(*), which is expected for N(s_(*), x_(*),m_(*)) at large s_(*). In contrast, the alternative expression with themodified Struve functions is well behaved and approaches δ(m_(*)−x_(*))at large s_(*).

Using the alternative expression for N(s_(*), x_(*), m_(*)), ananalytical expression for M(s_(*), x_(*)) consisting of specialfunctions such as the hypergeometric functions can also be obtained. Itis worth noting that evaluation of function M(s_(*), x_(*)) throughdirect numerical integration of the alternative expression for N(s_(*),x_(*), m_(*)) is also effective as long as sufficient integration pointsare added in the vicinity of the peak in N(s_(*), x_(*), m_(*)).

Finally, to solve for θ ₁(s_(*), x_(*)) in Eq. 12, a method ofsuccessive substitution can be adopted. Compared with the quadraturemethod, which was used in the previous literature and known in the art,the method of successive substitution overcomes the in-accuracy issue atlarge x_(*).

After solving the field variables in the Laplace domain, the solutionsin the time domain can be obtained by using the numerical inversionalgorithm of Stehfest, found in Stefest, H. 1970. “Algorithm 368:Numerical inversion of Laplace transforms”. Commun. ACM. 13(1): 47-49.

The normal stress on the contact surface (z=0) can be expressed as,

$\begin{matrix}{{\sigma_{z} = {\frac{2{G\left( {{2\eta} - 1} \right)}}{\pi \; \eta}{\frac{a}{R}\begin{bmatrix}{{2\left( {1 - r_{*}^{2}} \right)^{\frac{1}{2}}} + {{\theta_{1}\left( {t_{*},1} \right)}\left( {1 - r_{*}^{2}} \right)^{- \frac{1}{2}}} -} \\{\int_{r_{*}}^{1}{{\theta_{3}\left( {t_{*},x_{*}} \right)}\left( {x_{*}^{2} - r_{*}^{2}} \right)^{- \frac{1}{2}}{dx}_{*}}}\end{bmatrix}}}}{{where},{{\theta_{1}\left( {t_{*},x_{*}} \right)} = {\mathcal{L}^{- 1}\left\lbrack {s_{*}^{- 1}x_{*}^{\frac{1}{2}}{{\overset{¯}{\theta}}_{1}\left( {s_{*},x_{*}} \right)}} \right\rbrack}}}{{\theta_{3}\left( {t_{*},x_{*}} \right)} = \frac{\partial{\theta_{1}\left( {t_{*},x_{*}} \right)}}{\partial x_{*}}}} & (13)\end{matrix}$

and

⁻¹ is the Laplace inversion operator.

It is interesting to note that the ratio between the small- andlarge-time asymptotes for σ_(z)=0 is,

$\begin{matrix}{\left. \frac{\lim\limits_{t\rightarrow 0}\sigma_{z}}{\lim\limits_{t\rightarrow\infty}\sigma_{z}} \right|_{z = 0} = {\frac{2\eta}{\left( {{2\eta} - 1} \right)\varphi} = {1 + \omega}}} & (14)\end{matrix}$

Since the contact radius can be assumed to remain fixed in this model,the ratio of the contact pressure is the same as the ratio of theindentation force between the small and large times, namely,

$\begin{matrix}{\frac{F(0)}{F(\infty)} = {1 + \omega}} & (15)\end{matrix}$

where F(t) is the indentation force as a function of time. Indeed,integrating the normal stress over the contact area at t=0 and t→∞, theHertzian solutions for the indentation force are recovered,

${{F(0)} = \frac{16Ga^{3}}{3\varphi \; R}}{{F(\infty)} = \frac{8{G\left( {{2\eta} - 1} \right)}a^{3}}{3\eta R}}$

Variation of the contact pressure with time is shown in FIG. 3. Overallthe poroelastic solution is bounded by the Hertzian solutions at t=0 andt→∞. However, it can be seen from FIG. 3 that at intermediate times,σ_(z) is singular at the contact edge r_(*)=1. The assumption of a fixedcontact radius effectively allows discontinuity and singularity todevelop at the contact edge at intermediate times. Hence the solutiondeveloped in the present disclosure can be considered equivalent to thecase where there is a corner at the contact edge, but the tip of theindenter is spherical.

The results in FIG. 3 are obtained with the radius of the spherical-tipindenter taken as R=50 mm and the indentation depth d=0.1 mm, whichcorrespond to a contact radius of a=2.23 mm. Material properties of theGulf of Mexico Shale as listed in Cheng, A. H-D. 2016. Poroelasticity.Springer-Verlag. Berlin, can be used for this and subsequentcalculations. The saturating fluid can be assumed to be water ofviscosity μ=1 cp; skeleton shear modulus is 0.76 GPa; skeletoncompression modulus is 1.1 GPa; solid compression modulus is 34 GPa;fluid compression modulus is 2.25 GPa; porosity is 0.3 and permeabilityis 1e-19 m2.

Indentation force at an intermediate time can be expressed in anormalized form,

$\begin{matrix}{{F_{n}\left( t_{*} \right)} = \frac{{F\left( t_{*} \right)} - {F(\infty)}}{{F(0)} - {F(\infty)}}} & (16)\end{matrix}$

The explicit expression for F_(n)(t_(*)) is,

$\begin{matrix}{{F_{n}\left( t_{*} \right)} = {\frac{3}{2\omega}{\int_{0}^{1}{x_{*}^{1/2}{\mathcal{L}^{- 1}\left\lbrack {s_{*}^{- 1}{\theta_{1}\left( {s_{*},x_{*}} \right)}} \right\rbrack}{dx}_{*}}}}} & (17)\end{matrix}$

Eq. 17 shows that the normalized indentation force is a function ofconstant ω only. Note that the dimensionless time is defined ast_(*)=tc/a². In principle, the two force asymptotes determine thematerial constants G/ϕ, G(2η−1)/η and the ratio of the two forceasymptotes gives constant ω. Once ω is known, the force relaxationcurves in FIG. 6 can serve as the master curves to determine thediffusion coefficient c. However, it is noted that the force-relaxationcurve appears to be insensitive to w. This means that theforce-relaxation curve could be a rather reliable mean for determiningthe diffusivity coefficient c since the uncertainty in w does not have astrong effect on the force-relaxation behavior.

The solution procedures for solving Case II and III are similar to thatof Case I. The set of equations in Case II for unknowns A₁, A₂ and B₁is,

A ₁ ξ+A ₂√{square root over (ξ²+λ)}+B ₁(1−ϕ)=0

ηλA ₂√{square root over (ξ²+λ)}+[ϕ+2η(1−ϕ)]B ₁ξ²=0

The Fredholm integral equation for θ₁(s_(*), x_(*)) is,

θ₁(s_(*), x_(*)) + ω∫₁^(∞)N(s_(*), x_(*), m_(*))θ₁(s_(*), m_(*))dm_(*) = ωM(s_(*), x_(*))${where},{{N\left( {s_{*},x_{*},m_{*}} \right)} = {m_{*}{\int_{0}^{\infty}{\xi_{*}{H_{1}\left( {s_{*},\xi_{*}} \right)}{J_{- \frac{1}{2}}\left( {x_{*}\xi_{*}} \right)}{J_{- \frac{1}{2}}\left( {m_{*}\xi_{*}} \right)}d\; \xi_{*}}}}}$${M\left( {s_{*},x_{*}} \right)} = {\int_{0}^{1}{{m_{*}^{- \frac{1}{2}}\left( {1 - m_{*}^{2}} \right)}{N\left( {s_{*},x_{*},m_{*}} \right)}dm_{*}}}$${H_{1}\left( {s_{*},\xi_{*}} \right)} = {1 - {2s_{*}^{- 1}\xi_{*}^{2}} + {2s_{*}^{- 1}{\xi_{*}^{3}\left( {\xi_{*}^{2} + s_{*}} \right)}^{- \frac{1}{2}}}}$

Again B₁ and θ₁ are connected through an integral.

For Case III, the set of equation now includes one algebraic equationand two Fredholm integral equations,

$\mspace{20mu} {{{{A_{1}\xi} + {A_{2}\sqrt{\xi^{2} + \lambda}} + {B_{1}\left( {1 - \varphi} \right)}} = {0\mspace{14mu} {and}}},{{{\theta_{1\; a}\left( {s_{*},x_{*}} \right)} + {\omega {\int_{1}^{\infty}{{N_{a}\left( {s_{*},x_{*},m} \right)}{\theta_{1a}\left( {s_{*},m_{*}} \right)}d\; m_{*}}}} + {\omega {\int_{1}^{\infty}{{N_{b}\left( {s_{*},x_{*},m} \right)}{\theta_{1b}\left( {s_{*},m_{*}} \right)}d\; m_{*}}}}} = {{\omega \; {M_{a}\left( {s_{*},x_{*}} \right)}{{\theta_{1\; b}\left( {s_{*},x_{*}} \right)} + {\int_{1}^{\infty}{{N_{d}\left( {s_{*},x_{*},m} \right)}{\theta_{1a}\left( {s_{*},m_{*}} \right)}d\; m_{*}}} - {\int_{1}^{\infty}{{N_{c}\left( {s_{*},x_{*},m} \right)}{\theta_{1b}\left( {s_{*},m_{*}} \right)}d\; m_{*}}}}} = {{M_{b}\left( {s_{*},x_{*}} \right)}\mspace{14mu} {where}}}},\mspace{20mu} {N_{a} = {{m_{*}{\int_{0}^{\infty}{\xi_{*}{H_{1a}\left( {s_{*},\xi_{*}} \right)}{J_{- \frac{1}{2}}\left( {x_{*}\xi_{*}} \right)}{J_{- \frac{1}{2}}\left( {m_{*}\xi_{*}} \right)}d\; \xi_{*}\mspace{20mu} N_{b}}}} = {{2s_{*}^{- \frac{1}{2}}m_{*}{\int_{0}^{\infty}{\xi_{*}^{2}{H_{1b}\left( {s_{*},\xi_{*}} \right)}{J_{- \frac{1}{2}}\left( {x_{*}\xi_{*}} \right)}{J_{\frac{1}{2}}\left( {m_{*}\xi_{*}} \right)}d\; \xi_{*}\mspace{20mu} N_{c}}}} = {{m_{*}{\int_{0}^{\infty}{\xi_{*}{H_{1b}\left( {s_{*},\xi_{*}} \right)}{J_{\frac{1}{2}}\left( {x_{*}\xi_{*}} \right)}{J_{\frac{1}{2}}\left( {m_{*}\xi_{*}} \right)}d\; \xi_{*}\mspace{20mu} N_{d}}}} = {{s_{*}^{- \frac{1}{2}}m_{*}{\int_{0}^{\infty}{\xi_{*}^{2}{H_{1b}\left( {s_{*},\xi_{*}} \right)}{J_{\frac{1}{2}}\left( {x_{*}\xi_{*}} \right)}{J_{- \frac{1}{2}}\left( {m_{*}\xi_{*}} \right)}d\; \xi_{*}\mspace{20mu} M_{a}}}} = {\left( {s_{*},x_{*}} \right) = {{\int_{0}^{1}{{m_{*}^{- \frac{1}{2}}\left( {1 - m_{*}^{2}} \right)}{N_{a}\left( {s_{*},x_{*},m_{*}} \right)}d\; m_{*}\mspace{20mu} M_{b}}} = {\left( {s_{*},x_{*}} \right) = {{\int_{0}^{1}{{m_{*}^{- \frac{1}{2}}\left( {1 - m_{*}^{2}} \right)}{N_{d}\left( {s_{*},x_{*},m_{*}} \right)}d\; m_{*}\mspace{20mu} {H_{1a}\left( {s_{*},\xi_{*}} \right)}}} = {{1 - {2s_{*}^{- 1}\xi_{*}^{2}} + {2s_{*}^{- 1}{\xi_{*}^{3}\left( {\xi_{*}^{2} + s_{*}} \right)}^{- \frac{1}{2}}\mspace{20mu} {H_{1b}\left( {s_{*},\xi_{*}} \right)}}} = {1 - {\xi_{*}\left( {\xi_{*}^{2} + s_{*}} \right)}^{- \frac{1}{2}}}}}}}}}}}}}}$

Eq. 17 can also be used to calculate the normalized indentation forcefor Case II. Replacing θ₁(s_(*), x_(*)) with θ_(1a)(s_(*), x_(*)) givesthe normalized force relaxation expression for Case III. Summary of thenormalized indentation force relaxation for all three cases is shown inFIG. 6.

Variation of the radial stress with depth at intermediate time is infact non-monotonic, see FIG. 4. Since a drained boundary is assumed onthe surface, the radial stress on the surface drops almostinstantaneously from the undrained asymptote to a value slightly belowthe drained asymptote. Meanwhile, the pore pressure distribution alongthe depth shows the Mandel-Cryer effect, where the pore pressure risesabove the initial value at t_(*)=0⁺ before its dissipation, as shown inFIG. 5.

Reference will now be made in detail to exemplary embodiments of thedisclosed technology, examples of which are illustrated in theaccompanying drawings and disclosed herein. Wherever convenient, thesame references numbers will be used throughout the drawings to refer tothe same or like parts.

FIGS. 1-2 and 10 illustrate exemplary embodiments of the presentlydisclosed systems and methods for characterizing poroelastic materials.

In FIG. 1, a method 100 for characterizing poroelastic materials isdisclosed herein. In block 110, experimental data can be obtained from amaterial test, comprising at least time data and indentation force data.In some embodiments, the experimental data can be obtained using anindentation tool configured to indent a material to a predetermineddepth. The indentation tool can comprise a spherical indenter in theform of a rigid sphere. For example, the indenter can be a rigid smoothsphere. Additionally, the state of the indenter can be permeable orimpermeable.

In some embodiments, the experimental data can be obtained through oneor more force sensors (e.g., two or more, three or more, four or more,or five or more). For example, a force sensor can be housed in theindentation tool to measure the force applied to the material.Additional force sensors can be attached to the system in anyconfiguration such that the applied force to the material can beobtained from the sensors. In some embodiments, the force sensors can beconnected to one or more storage devices (e.g., two or more, three ormore, four or more, or five or more). Suitable examples of a storagedevice can include, but are not limited to, hard drives, hard disks,solid-state drives, removable universal serial bus (USB) drives, floppydisks, compact disks (CDs) and the like. The one or more storage devicescan be configured to store the experimental data, along with other data.It is understood that the storage devices can store more than theexperimental data.

In some embodiments, the one or more force sensors, the indentationtool, and/or the one or more storage devices can be connected to one ormore processors (e.g., two or more, three or more, four or more, or fiveor more). The one or more processors can be configured to executeinstructions given to the system. For example, the one or moreprocessors can be configured to cause the indentation tool to indent thematerial and can cause the one or more storage devices to beginrecording the experimental data received from the one or more forcesensors. Additionally, the system can comprise at least one memoryconfigured to store instructions. In some embodiments, the instructionsstored on the at least one memory can be executed by the one or moreprocessors. For example, the presently disclosed method steps can bestored in the memory and executed by the one or more processors.

In block 120, a first asymptote of the indentation force data can beindicated with a corresponding first time from the time data. In someembodiments, the first time from the time data can be the initial timeand correspond with an initial indentation force asymptote. As would beappreciated by one of ordinary skill in the art, the first asymptote ofthe indentation force data would be the initial force required to indenta material to a predetermined depth. In some embodiments, the indicationcan be received from the one or more processors. Additionally, theindication can be stored on the one or more storage devices along withthe relevant portions of the experimental data.

In block 130, a second asymptote of the indentation force data can beindicated with a corresponding second time from the time data. In someembodiments, the second time from the time data can be the ending time,or termination time of the test. In other words, the second time can bethe time tending towards infinity and corresponding with a steady-stateindentation force asymptote. As would be appreciated by one of ordinaryskill in the art, as force relaxation occurs to hold the indentationtool at the predetermined depth, and as time tends towards infinity, theforce required to maintain the tool at the predetermined depth willlevel-out or reach a steady-state value. This steady-state force valuecan be taken as the second asymptote. In some embodiments, theindication can be received from the one or more processors.Additionally, the indication can be stored on the one or more storagedevices along with the relevant portions of the experimental data.

In block 140, the ratio between the first and the second asymptotes canbe calculated and used to select a corresponding master curve from aplurality of master curves. In some embodiments, the ratio between thefirst and the second asymptotes can be calculated as a constant, whereeach master curve from the plurality of master curves corresponds to acertain value of the constant. As mentioned above, the ratio of theasymptotes can be the ratio between the initial force required tomaintain the indentation depth and the steady-state force required tomaintain the indentation depth. In some embodiments, the plurality ofmaster curves can be stored in the one or more storage devices. Thecalculation of the ratio can be performed by the one or more processors,which can then retrieve the correct corresponding master curve from theone or more storage devices.

In block 150, the selected master curve can be matched with theexperimental data to obtain desired material properties. There existmany data regression techniques which can be used to match theexperimental data to the selected master curve. In some embodiments, thedata matching can be performed by the one or more processors. Once thedata matching has been completed, the desired material properties can becalculated for the performed material test. For example, the desiredmaterial property can be the coefficient of hydraulic diffusion,hydraulic diffusivity (or coefficient of consolidation for soils), thehardness, the material toughness, and the like. In some embodiments, thecalculating can be performed by the one or more processors. Further, thecalculated material property values can be stored in the one or morestorage devices for later use.

In FIG. 2, a method 200 for constructing master curves for a poroelasticmaterial is disclosed herein. In some embodiments, the master curveconstruction can be carried out by the one or more processors.Additionally, the one or more processors can receive data input by auser based on the material being tested, such as material properties. Insome embodiments, the one or more processors can store the master curvesin one or more storage devices after construction. In block 210, one ormore governing equations and one or more boundary conditions can beestablished. In block 220, the boundary conditions and the governingequations can be transformed into the Laplace domain. In block 230,alternate integral expressions can be provided for the integral kernelsusing one or more modified Struve functions. In block 240, the integralequations can be solved using a method of successive substitution. Inblock 250, a time domain solution can be calculated by numericallyinverting the solved integral equations from the Laplace domain. Inblock 260, the time domain solutions can be integrated to obtain theplurality of master curves. In some embodiments, the master curves canbe normalized to make time dimensionless, and/or to normalize the forcebetween 0 and 1. Suitable examples of calculated master curves can beseen in FIG. 6.

In FIG. 7, a method 700 for characterizing poroelastic materials isdisclosed herein. In block 710, the indentation force can be recorded asa function of time. In some embodiments, the indentation force can forma force relaxation curve. In block 720, the early and late timeasymptotes can be denoted. The ratio between the asymptotes can becalculated to obtain a constant. In block 730, based on the constant, acorresponding master curve can be chosen to model the behavior of thematerial. In block 740, the indentation force can be normalized to bewithin 0 and 1, inclusive. In block 750, the experimental data can bematched with the corresponding master curve through data regression toobtain the desired material properties.

While the present disclosure has been described in connection with aplurality of exemplary aspects, as illustrated in the various figuresand discussed above, it is understood that other similar aspects can beused or modifications and additions can be made to the described aspectsfor performing the same function of the present disclosure withoutdeviating therefrom. For example, in various aspects of the disclosure,methods and compositions were described according to aspects of thepresently disclosed subject matter. However, other equivalent methods orcomposition to these described aspects are also contemplated by theteachings herein. Therefore, the present disclosure should not belimited to any single aspect, but rather construed in breadth and scopein accordance with the appended claims.

1. A method for characterizing a material comprising: selecting acorresponding master curve from a set of theoretical master curves basedon a ratio of a first asymptote of indentation force data and a secondasymptote of the indentation force data: wherein the indentation forcedata and time data are based upon a response of a material to anindentation force over time; wherein the first asymptote corresponds toa first time from the time data; and wherein the second asymptotecorresponds to a second time from the time data; and calculating a valuefor a property of the material by correlating the indentation force dataand time data with the corresponding master curve.
 2. The method ofclaim 1 further comprising obtaining the indentation force data and timedata for the material: wherein the material is a poroelastic material;and wherein obtaining the indentation force data and time datacomprises: indenting over time, with an indentation tool, theporoelastic material to a predetermined indentation depth; and measuringa force required to maintain the indentation tool at the predeterminedindentation depth.
 3. (canceled)
 4. The method of claim 2 wherein theindentation tool is selected from the group consisting of: permeablerigid sDhere indenters and impermeable rgi sphere indenters.
 5. Themethod of claim 2, wherein a drainage condition on a surface of theporoelastic material is selected from the group consisting of: apermeable indenter on a fully permeable surface, a permeable/impermeableindenter on a fully impermeable surface, and an impermeable indenter ona fully permeable surface.
 6. The method of claim 1, wherein the firsttime from the time data is an initial time; wherein the second time fromthe time data is an ending time; and wherein the indentation forcereaches a maximum.
 7. (canceled)
 8. The method of claim 1, wherein theproperty of the material is the coefficient of hydraulic diffusion. 9.(canceled)
 10. The method of claim 1, wherein selecting thecorresponding master curve comprises: calculating a constant comprisingthe ratio of the first and the second asymptotes; and selecting thecorresponding master curve from a theoretical solution corresponding tothe value of the constant and a drainage condition on a surface of thematerial.
 11. (canceled)
 12. A method for characterizing poroelasticmaterials comprising: indicating a first asymptote of indentation forcedata of experimental data for a material at a corresponding first timefrom time data of the experimental data for the material; indicating asecond asymptote of the indentation force data at a corresponding secondtime from the time data; selecting a corresponding master curve from aplurality of theoretical master curves based on the ratio of the firstand the second asymptotes; and calculating a value for a property of thematerial by matching the experimental data with the corresponding mastercurve; wherein the plurality of master curves is obtained by:transforming one or more governing equations using a Hankel transform inthe Laplace domain to find expressions for displacement functions interms of integrals with a number of unknowns; transforming one or moreboundary conditions to the Laplace domain and matching the boundaryconditions with field quantities expressed through the displacementfunctions to obtain an equation for each of the number of unknowns,which include one or more Fredholm integral equations of the secondkind; providing alternate integral expressions for integral kernels inthe one or more Fredholm integral equations using one or more modifiedStruve functions; solving the one or more Fredholm integral equationsusing a method of successive substitution; calculating a time domainsolution by numerically inverting the solved integral equations in theLaplace domain; and integrating, based on a plurality of parameters of aspecific poroelastic material, the time domain solution to obtain theplurality of master curves representing force relaxation.
 13. The methodof claim 12, wherein at least one of the governing equations is obtainedfrom a McNamee-Gibson displacement function method.
 14. The method ofclaim 12, wherein at least one of the governing equations comprises oneor more material constants.
 15. The method of claim 12, wherein one ormore field quantities expressed comprise the expressions for thedisplacement functions.
 16. A system for characterizing poroelasticmaterials comprising: an indentation tool comprising a rigid smoothsphere; one or more force sensors; one or more processors; and at leastone memory storing instructions that when executed by the one or moreprocessors, cause the system to: indent, with the indentation tool, aporoelastic material to a predetermined indentation depth; measure,using the one or more force sensors, a force required to maintain theindentation tool at the predetermined indentation depth to obtainindentation force data; record experimental data for the poroelasticmaterial, the experimental data comprising at least: indentation forcedata; and time data corresponding to the indentation force data;indicate a first asymptote of the indentation force data at acorresponding first time from the time data; indicate a second asymptoteof the indentation force data at a corresponding second time from thetime data; select a corresponding master curve from a plurality oftheoretical master curves based on the ratio of the first and the secondasymptotes; and calculate a value for a property of the poroelasticmaterial by matching the experimental data with the corresponding mastercurve.
 17. (canceled)
 18. The method of claim 16, wherein the rigidsmooth sphere is selected from the group consisting of: permeableindenters and impermeable indenters.
 19. The method of claim 16, whereinthe drainage condition on a surface of the poroelastic material isselected from the group consisting of: a permeable indenter on a fullypermeable surface, a permeable/impermeable indenter on a fullyimpermeable surface, and an impermeable indenter on a fully permeablesurface. 20.-22. (canceled)
 23. The system of claim 16, wherein theexperimental data further comprises at least a force relaxationbehavior.
 24. The system of claim 16, wherein selecting thecorresponding master curve comprises: calculating a constant comprisingthe ratio of the first and the second asymptotes; and selecting thecorresponding master curve from a theoretical solution corresponding tothe value of the constant and the drainage condition on a surface of theporoelastic material.
 25. The system of claim 16, wherein calculatingthe value of the property of the poroelastic material comprises fitting,using a fitting function predetermined by the full poroelastic solution,the experimental data to the corresponding master curve to obtain thevalue of the property of the poroelastic material.
 26. The system ofclaim 16, wherein the plurality of master curves is obtained by storinginstructions that when executed by the one or more processors, cause thesystem to: establish one or more governing equations; establish one ormore boundary conditions using Heaviside step displacement loading;transform the one or more governing equations using Hankel transform inthe Laplace domain to find the expressions for the two displacementfunctions in terms of integrals with three unknowns; transform the oneor more boundary conditions to the Laplace domain and matching theboundary conditions with the field quantities expressed through thedisplacement functions to obtain three equations for the three unknowns,which include one or more Fredholm integral equations of the secondkind; provide alternate integral expressions for the integral kernels inthe Fredholm integral equations using one or more modified Struvefunctions; solve the Fredholm integral equations using a method ofsuccessive substitution; calculate a time domain solution by numericallyinverting the solved Laplace domain equation; and integrate, based on aplurality of parameters of a specific poroelastic material, the timedomain solution to obtain a plurality of master curves representingforce relaxation.
 27. The system of claim 26, wherein the one or moregoverning equations are obtained from one or more McNamee-Gibsondisplacement function methods.
 28. The system of claim 26, wherein theone or more governing equations comprise one or more material constants.29. The system of claim 26, wherein the expressions of one or more fieldquantities comprise the displacement function solutions from the one ormore governing equations.